A239550 -id:A239550 - cp's OEIS frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.

1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 4, 3, 2, 3, 2, 2, 4, 3, 2, 3, 3, 2, 3, 4, 2, 3, 2, 2, 3, 3, 3, 5, 2, 2, 3, 3, 2, 3, 2, 2, 5, 3, 2, 3, 3, 2, 4, 3, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 3, 5, 2, 2, 3, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 2, 5, 3, 2, 3, 3, 2, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 2, 2, 3, 4, 2, 3, 2, 2, 5, 3

Offset: 2

Alexander Adamchuk, Apr 27 2007

nonn

The indices k of the first appearance of number n in a(k) are listed in A063778(n) = {2,3,6,15,36,225,...} = Least number k>1 such that k could be represented in n different ways as general m-gonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)).
From Gus Wiseman, May 03 2019: (Start)
Also the number of integer partitions of n whose augmented differences are all equal, where the augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k; for example aug(6,5,5,3,3,3) = (2,1,3,1,1,3). Equivalently, a(n) is the number of integer partitions of n whose differences are all equal to the last part minus one. The Heinz numbers of these partitions are given by A307824. For example, the a(35) = 5 partitions are:
  (35)
  (23,12)
  (11,9,7,5,3)
  (8,7,6,5,4,3,2)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
(End)

			a(6) = 3 because 6 = P(2,6) = P(3,3) = P(6,2).

Alois P. Heinz, Table of n, a(n) for n = 2..10000
E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A063778, A177025.
Column k=0 of A239550.
Cf. A007862, A049988, A307824, A325349, A325350, A325356, A325357, A325358, A325458, A325459.

A129654 := proc(n) local resul, dvs, i, r, m ;
   dvs := numtheory[divisors](2*n) ;
   resul := 0 ;
   for i from 1 to nops(dvs) do
      r := op(i, dvs) ;
      if r > 1 then
         m := (2*n/r-4+2*r)/(r-1) ;
         if is(m, integer) then
            resul := resul+1 ;
         fi ;
      fi ;
   od ;
   RETURN(resul) ;
end: # R. J. Mathar, May 14 2007

a[n_] := (dvs = Divisors[2*n]; resul = 0; For[i = 1, i <= Length[dvs], i++, r = dvs[[i]]; If[r > 1, m = (2*n/r-4+2*r)/(r-1); If[IntegerQ[m], resul = resul+1 ] ] ]; resul); Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Sep 13 2012, translated from R. J. Mathar's Maple program *)
Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]], {n, 2, 106}] (* Jonathan Sondow, May 09 2014 *)
atpms[n_]:=Select[Join@@Table[i*Range[k,1,-1],{k,n},{i,0,n}],Total[#+1]==n&];
Table[Length[atpms[n]],{n,100}] (* Gus Wiseman, May 03 2019 *)

a(n) = sumdiv(2*n, d, (d>1) && (2*n/d + 2*d - 4) % (d-1) == 0); \\ Daniel Suteu, Dec 22 2018

a(n) = A177025(n) + 1.

G.f.: x * Sum_{k>=1} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020

A239551 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-1,0,1}.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 24, 34, 49, 71, 100, 145, 205, 295, 423, 610, 872, 1260, 1804, 2599, 3733, 5381, 7725, 11131, 15996, 23042, 33133, 47714, 68608, 98793, 142101, 204585, 294318, 423683, 609565, 877457, 1262508, 1817205, 2614832, 3763553, 5415668

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

			a(6) = 6: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [1,2,2,1], [1,2,3], [1,5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=1 of A239550.

a(n) ~ c * d^n, where d=1.4391340589699362028978918824612984596732704665024595117321768607966..., c=0.5987103268327131789103863373328359387911710658212178254615152936898... - Vaclav Kotesovec, May 01 2014

A239552 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-2,...,2}.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 11, 18, 32, 53, 89, 152, 255, 431, 733, 1243, 2111, 3583, 6087, 10342, 17564, 29845, 50704, 86150, 146382, 248716, 422618, 718091, 1220156, 2073272, 3522846, 5985967, 10171232, 17282786, 29366645, 49899299, 84788109, 144070585, 244802450

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

			a(5) = 7: [1,1,1,1,1], [1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,4].
a(6) = 11: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,1,2,1], [1,1,1,3], [1,1,2,1,1], [1,1,2,2], [1,2,1,1,1], [1,2,1,2], [1,2,2,1], [1,2,3], [1,5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=2 of A239550.

a(n) ~ c * d^n, where d = 1.6991842071646453928920987915033839182675484195426953647249305754952..., c = 0.4361924842463814839782496098129122542027259655263562779280217228428... . - Vaclav Kotesovec, Aug 28 2014

A239553 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-3,...,3}.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 23, 40, 73, 132, 235, 425, 767, 1379, 2491, 4495, 8105, 14629, 26407, 47647, 86009, 155251, 280202, 505771, 912930, 1647796, 2974292, 5368610, 9690288, 17491016, 31571338, 56986216, 102860299, 185663064, 335122028, 604896038, 1091838582

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=3 of A239550.

a(n) ~ c * d^n, where d = 1.80500192473773030222699209897983267121107478038035391729605116..., c = 0.35357751116733859348564640393076257358506245520421137739865819... . - Vaclav Kotesovec, Aug 28 2014

A239554 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-4,...,4}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 28, 52, 97, 183, 344, 647, 1219, 2295, 4322, 8140, 15338, 28899, 54449, 102588, 193297, 364219, 686272, 1293113, 2436562, 4591090, 8650808, 16300370, 30714116, 57873375, 109048498, 205475699, 387169634, 729528290, 1374621010, 2590143467

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=4 of A239550.

A239555 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-5,...,5}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 29, 56, 106, 201, 388, 743, 1421, 2727, 5233, 10034, 19248, 36925, 70836, 135885, 260684, 500101, 959400, 1840522, 3530904, 6773757, 12994934, 24929791, 47825908, 91750372, 176016138, 337673592, 647801215, 1242757517, 2384136115

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A239550.

A239556 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-6,...,6}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 31, 60, 117, 228, 441, 860, 1678, 3267, 6368, 12413, 24192, 47158, 91929, 179190, 349306, 680925, 1327341, 2587450, 5043845, 9832184, 19166364, 37361945, 72831417, 141973854, 276756612, 539495115, 1051664254, 2050060645, 3996283405

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A239550.

A239557 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-7,...,7}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 31, 61, 120, 235, 460, 901, 1769, 3471, 6811, 13367, 26236, 51497, 101079, 198399, 389428, 764384, 1500372, 2945008, 5780612, 11346484, 22271478, 43715623, 85807323, 168427136, 330597667, 648914509, 1273723636, 2500131931, 4907390780

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=7 of A239550.

A239560 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-10,...,10}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 64, 127, 252, 501, 997, 1982, 3940, 7832, 15575, 30972, 61590, 122481, 243567, 484363, 963221, 1915501, 3809243, 7575217, 15064391, 29957679, 59575090, 118473530, 235601447, 468526933, 931732343, 1852882096, 3684719208, 7327587468

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=10 of A239550.

A239561 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-n,...,n}.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 31, 63, 125, 252, 504, 1013, 2027, 4069, 8141, 16318, 32650, 65381, 130801, 261791, 523677, 1047780, 2095796, 4192533, 8385623, 16773321, 33547917, 67100362, 134203614, 268417029, 536840509, 1073702131, 2147418493, 4294882224, 8589795592

Offset: 0

Alois P. Heinz, Mar 21 2014

nonn

			There are 2^5 = 32 compositions of 7 with first part = 1.  Exactly one of these has second differences not in {-7,...,7}, namely [1,5,1].  Thus a(7) = 32 - 1 = 31.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Main diagonal of A239550.

b:= proc(n) option remember; `if`(n<5, [1, 1, 3, 4, 8][n+1],
      (-(n^3+3*n^2+184*n-348) *b(n-1)
       +(2*n^4+23*n^3-155*n^2-166*n+3776) *b(n-2)
       +(n^4+14*n^3-5*n^2+122*n+768) *b(n-3)
       +(2*n^3+10*n^2-64*n-1328) *b(n-4)
       -(2*n^4+28*n^3-78*n^2-272*n+2320) *b(n-5))/
      (n^4+10*n^3-75*n^2-20*n+1244))
    end:
a:= n-> `if`(n<7, ceil(2^(n-2)), 2^(n-2)-b(n-7)):
seq(a(n), n=0..40);

b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n - h, j, h, k], {h, 1, n}], Sum[b[n - h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]];
a[n_] := If[n == 0, 1, b[n - 1, 0, 1, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

a(n) ~ 2^(n-2). - Vaclav Kotesovec, May 01 2014

cp's OEIS Frontend

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2*r*((m-2)*r-(m-4)) = n>1, for m,r>1.

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A239551 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-1,0,1}.

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A239552 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-2,...,2}.

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A239553 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-3,...,3}.

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A239554 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-4,...,4}.

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A239555 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-5,...,5}.

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A239556 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-6,...,6}.

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A239557 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-7,...,7}.

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A239560 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-10,...,10}.

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A239561 Number of compositions of n such that the first part is 1 and the second differences of the parts are in {-n,...,n}.

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Author

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Programs

Maple

Mathematica

Formula

A129654 Number of different ways to represent n as general polygonal number P(m,r) = 1/2r((m-2)*r-(m-4)) = n>1, for m,r>1.