A079826 -id:A079826 - cp's OEIS frontend

A079825 Sum of numbers in n-th upward diagonal of triangle in A079826.

1, 3, 6, 15, 26, 41, 62, 92, 129, 169, 224, 287, 366, 443, 548, 656, 793, 919, 1090, 1255, 1466, 1653, 1906, 2140, 2441, 2701, 3052, 3367, 3774, 4119, 4584, 4992, 5521, 5963, 6558, 7071, 7738, 8289, 9030, 9660, 10481, 11153, 12056, 12815, 13806, 14611, 15692, 16592

Offset: 1

Amarnath Murthy, Feb 11 2003

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,2,-2,-2,2,-1,1,1,-1).

Cf. A056011, A079826.

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8)/((x^2+1)^2*(1+x)^3*(x-1)^4))); // Bruno Berselli, Sep 05 2012

[((2*n+1)*(7*n^2+7*n+12) -3*(-1)^n*n*(n-7) +6*(-1)^Binomial(n,2)*(2*n-3*(-1)^n+1))/96: n in [1..41]]; // G. C. Greubel, Dec 10 2023

A079825 := proc(n)
        local a, k;
        a := 0 ;
        for k from 1 to (n+1)/2 do
                a := a+A056011(n-k+1,k) ;
        end do:
        a ;
end proc: # R. J. Mathar, Sep 05 2012

LinearRecurrence[{1, 1, -1, 2, -2, -2, 2, -1, 1, 1, -1}, {1, 3, 6, 15, 26, 41, 62, 92, 129, 169, 224}, 35] (* Bruno Berselli, Sep 05 2012 *)

makelist(expand(((2*n+1)*(7*n^2+7*n+12)-3*n*(n-7)*(-1)^n+6*(2*n-3*(-1)^n+1)*%i^(n*(n-1)))/96), n, 1, 35); /* Bruno Berselli, Sep 05 2012 */

[((2*n+1)*(7*n^2+7*n+12)-3*(-1)^n*n*(n-7)+6*(-1)^binomial( n,2)*(2*n-3*(-1)^n+1))/96 for n in range(1,41)] # G. C. Greubel, Dec 10 2023

G.f.: x*( 1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8 ) / ( (1+x)^3*(1-x)^4*(1+x^2)^2 ). - R. J. Mathar, Sep 05 2012
a(n) = ( (2*n+1)*(7*n^2+7*n+12) -3*n*(n-7)*(-1)^n +6*(2*n-3*(-1)^n+1)*i^(n*(n-1)) )/96, where i=sqrt(-1). - Bruno Berselli, Sep 05 2012
E.g.f.: (1/48)*( (6+24*x+30*x^2+7*x^3)*cosh(x) + (6+42*x+33*x^2+7*x^3)* sinh(x) + 6*(x-1)*cos(x) - 6*(x-2)*sin(x) ). - G. C. Greubel, Dec 10 2023

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 10, 9, 8, 7, 11, 12, 13, 14, 15, 21, 20, 19, 18, 17, 16, 22, 23, 24, 25, 26, 27, 28, 36, 35, 34, 33, 32, 31, 30, 29, 37, 38, 39, 40, 41, 42, 43, 44, 45, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Clark Kimberling, Aug 01 2000

A triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.
Self-inverse permutation of the natural numbers.
Mirror image of triangle in A056023. - Philippe Deléham, Apr 04 2009
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012
For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as a table:
   1,  3,  4, 10, 11, 21, ...
   2,  5,  9, 12, 20, 23, ...
   6,  8, 13, 19, 24, 34, ...
   7, 14, 18, 25, 33, 40, ...
  15, 17, 26, 32, 41, 51, ...
  ...
Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
The start of the sequence as triangle array read by rows:
   1;
   3,  2;
   4,  5,  6;
  10,  9,  8,  7;
  11, 12, 13, 14, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A079826, A131179 (first column), A218890, A213927.

Cf. A000027, A038722.

a056011 n = a056011_tabl !! (n-1)
a056011_list = concat a056011_tabl
a056011_tabl = ox False a000027_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
a056011_row n = a056011_tabl !! (n-1)
-- Reinhard Zumkeller, Nov 08 2013

A056011 := proc(n,k)
        if type(n,'even') then
                A131179(n)-k+1 ;
        else
                A131179(n)+k-1 ;
        end if;
end proc: # R. J. Mathar, Sep 05 2012

Flatten[If[EvenQ[Length[#]],Reverse[#],#]&/@Table[c=(n(n+1))/2;Range[ c-n+1,c],{n,20}]] (* Harvey P. Dale, Mar 25 2012 *)
With[{nn=20},{#[[1]],Reverse[#[[2]]]}&/@Partition[ TakeList[ Range[ (nn(nn+1))/2],Range[nn]],2]//Flatten] (* Harvey P. Dale, Oct 05 2021 *)

a(n) = ((i+j-1)*(i+j-2)+((-1)^t+1)*i - ((-1)^t-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

New name from Peter Luschny, Apr 15 2023, based on Boris Putievskiy's comment

cp's OEIS Frontend

A079825 Sum of numbers in n-th upward diagonal of triangle in A079826.

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