A005230 -id:A005230 - cp's OEIS frontend

A066777 First differences of Stern's sequence A005230.

0, 1, 1, 3, 5, 9, 20, 37, 71, 137, 285, 550, 1080, 2123, 4175, 8498, 16711, 33137, 65724, 130368, 258613, 521549, 1034600, 2060702, 4104693, 8176249, 16286774, 32443180, 65149296, 129777043, 259032537, 517030474, 1032000246, 2059895799, 4111615349

Offset: 0

N. J. A. Sloane, Jul 04 2002

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A005230.

A066777list[nmax_]:=Differences[Nest[Append[#,Total[Take[#,-Ceiling[(Sqrt[8Length[#]+1]-1)/2]]]]&,{1},nmax+1]];A066777list[35] (* Paolo Xausa, Sep 28 2023 *)

a(n) = A005230(n+2) - A005230(n+1). - Paolo Xausa, Sep 28 2023

A096858 Triangle read by rows in which row n gives the n-set obtained as the differences {b(n)-b(n-i), 0 <= i <= n-1}, where b() = A005318().

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 6, 9, 11, 12, 13, 11, 17, 20, 22, 23, 24, 20, 31, 37, 40, 42, 43, 44, 40, 60, 71, 77, 80, 82, 83, 84, 77, 117, 137, 148, 154, 157, 159, 160, 161, 148, 225, 265, 285, 296, 302, 305, 307, 308, 309, 285, 433, 510, 550, 570, 581, 587, 590, 592, 593, 594

Offset: 1

N. J. A. Sloane, Aug 18 2004

It is conjectured that the triangle has the property that all 2^n subsets of row n have distinct sums. This conjecture was proved by T. Bohman in 1996 - N. J. A. Sloane, Feb 09 2012

It is also conjectured that in some sense this triangle is optimal. See A005318 for further information and additional references.

			The triangle begins:
{1}
{1,2}
{2,3,4}
{3,5,6,7}
{6,9,11,12,13}
{11,17,20,22,23,24}
{20,31,37,40,42,43,44}
{40,60,71,77,80,82,83,84}
{77,117,137,148,154,157,159,160,161}
{148,225,265,285,296,302,305,307,308,309}
{285,433,510,550,570,581,587,590,592,593,594}
{570,855,1003,1080,1120,1140,1151,1157,1160,1162,1163,1164}
{1120,1690,1975,2123,2200,2240,2260,2271,2277,2280,2282,2283,2284}
{2200,3320,3890,4175,4323,4400,4440,4460,4471,4477,4480,4482,4483,4484}
{4323,6523,7643,8213,8498,8646,8723,8763,8783,8794,8800,8803,8805,8806,8807}

J. H. Conway and R. K. Guy, Solution of a problem of Erdős, Colloq. Math. 20 (1969), p. 307.
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
R. K. Guy, Unsolved Problems in Number Theory, C8.

Alois P. Heinz, Rows n = 1..141, flattened
Tom Bohman, A sum packing problem of Erdős and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.

Cf. A005318, A005230 (column 1 of triangle).

b:= proc(n) option remember;
      `if`(n<2, n, 2*b(n-1) -b(n-1-floor(1/2 +sqrt(2*n-2))))
    end:
T:= n-> seq(b(n)-b(n-i), i=1..n):
seq(T(n), n=1..15);  # Alois P. Heinz, Nov 29 2011

b[n_] := b[n] = If[n < 2, n, 2*b[n-1] - b[n-1-Floor[1/2 + Sqrt[2*n-2]]]]; t[n_] := Table[b[n] - b[n-i], {i, 1, n}]; Table[t[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Alois P. Heinz *)

Typo in definition (limits on i were wrong) corrected and reference added to Bohman's paper. N. J. A. Sloane, Feb 09 2012

A023536 Convolution of natural numbers with A023532.

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 25, 31, 38, 46, 55, 65, 75, 86, 98, 111, 125, 140, 155, 171, 188, 206, 225, 245, 266, 287, 309, 332, 356, 381, 407, 434, 462, 490, 519, 549, 580, 612, 645, 679, 714, 750, 786, 823, 861, 900, 940, 981, 1023, 1066, 1110, 1155

Offset: 1

Clark Kimberling

nonn

From Vladimir Letsko, Dec 18 2016: (Start)
Also, a(n) is the number of possible values for the number of diagonals in a convex polyhedron with n+3 vertices.
Let v>4 denote the number of vertices of convex polyhedra. The set of possible numbers of diagonals is the union of sets {(k-1)(v-k-4), ..., (k-1)(v-(k+6)/2)}, where 1 <= k <= floor((sqrt(8v-15)-5)/2), and the set {(k-1)(v-k-4), ..., (v-3)(v-4)/2}, where k = floor((sqrt(8v-15)-3)/2). Note that cardinalities of all sets of this union excluding the last one are consecutive triangular numbers. (End)

Vladimir Letsko, Table of n, a(n) for n = 1..500

Cf. A005230, A279681.

A023536[n_] := n*(n + 5)/2 - 2 - Sum[Round[Sqrt[2*k + 4]], {k, 2, n}];
Array[A023536, 60] (* Paolo Xausa, Feb 28 2025 *)

from math import comb, isqrt
def A023536(n): return comb(n+2,2)-sum(isqrt((k<<3)+1)-1>>1 for k in range(1,n+2)) # Chai Wah Wu, Feb 27 2025

a(n) = (n(n + 5) - 4 )/2 - Sum_{k=2..n} floor(1/2 + sqrt(2(k + 2))). - Jan Hagberg (jan.hagberg(AT)stat.su.se), Oct 16 2002
From Paul Barry, May 24 2004: (Start)
a(n) = (n+1)(n+2)/2 - Sum_{k=1..n+1} floor((sqrt(8k+1)-1)/2);
a(n) = Sum_{k=1..n+1} k-floor((sqrt(8k+1)-1)/2). (End)

Corrected by Jan Hagberg (jan.hagberg(AT)stat.su.se), Oct 16 2002

A225178 The generalized Conway-Guy sequence d_2(n).

Original entry on oeis.org

1, 2, 6, 16, 48, 140, 408, 1224, 3640, 10824, 32192, 96576, 288912, 864288, 2585584, 7735104, 23205312, 69551552, 208461504, 624806688, 1872691488, 5612903296, 16838709888, 50500659456, 151455567744, 454227600128, 1362265877376

Offset: 1

N. J. A. Sloane, May 02 2013

nonn

Jaegug Bae and Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n).

Cf. A005318, A005230, A002024.

b := proc(n)
    round(sqrt(2*n-2)) ;
end proc:
d := proc(k,n)
    option remember;
    if n = 1 then
        1;
    else
        add( k*procname(k,i),i=n-b(n)..n-1 ) ;
    end if;
end proc:
A225178 := proc(n)
    d(2,n) ;
end proc: # R. J. Mathar, Jul 09 2013

b[n_] := Round[Sqrt[2n-2]];
d[k_, n_] := d[k, n] = If[n == 1, 1, Sum[k*d[k, i], {i, n-b[n], n-1}]];
a[n_] := d[2, n];
Table[a[n], {n, 1, 27}] (* Jean-François Alcover, Feb 27 2024, after R. J. Mathar *)

Bae and Choi define this sequence via a collection of recurrences.

A014172 Apply partial sum operator twice to Stern's sequence.

Original entry on oeis.org

1, 3, 7, 14, 27, 51, 95, 179, 340, 649, 1243, 2407, 4691, 9175, 17982, 35287, 69588, 137596, 272448, 539868, 1070224, 2122129, 4217132, 8389833, 16700934, 33255128, 66228664, 131908316

Offset: 0

N. J. A. Sloane

nonn

Index entries for sequences related to Stern's sequences

Cf. A005230.

A014173 Apply partial sum operator thrice to Stern's sequence.

Original entry on oeis.org

1, 4, 11, 25, 52, 103, 198, 377, 717, 1366, 2609, 5016, 9707, 18882, 36864, 72151, 141739, 279335, 551783, 1091651, 2161875, 4284004, 8501136, 16890969, 33591903, 66847031, 133075695, 264984011

Offset: 0

N. J. A. Sloane

nonn

Index entries for sequences related to Stern's sequences

Cf. A005230.

A014175 Apply partial sum operator 4 times to Stern's sequence.

Original entry on oeis.org

1, 5, 16, 41, 93, 196, 394, 771, 1488, 2854, 5463, 10479, 20186, 39068, 75932, 148083, 289822, 569157, 1120940, 2212591, 4374466, 8658470, 17159606, 34050575, 67642478, 134489509, 267565204, 532549215

Offset: 0

N. J. A. Sloane

nonn

Index entries for sequences related to Stern's sequences

Cf. A005230.

A132156 Sum of all n-digit Stern numbers.

Original entry on oeis.org

13, 148, 1003, 16141, 117547, 917053, 15502289, 114274754, 909482399, 15471962269, 115061008452, 917885746619, 15675637954053, 116877859994048, 933643385166035, 15975162211603393, 119196492782053456, 952870691477908028, 16319572144410372393, 121792577413939691253

Offset: 1

Parthasarathy Nambi, Nov 01 2007

			Sum of all 1-digit Stern numbers is 1 + 1 + 2 + 3 + 6 = 13.
Sum of all 2-digit Stern numbers is 11 + 20 + 40 + 77 = 148.
Sum of all 3-digit Stern numbers is 148 + 285 + 570 = 1003.

Cf. A005230.

stern[1] = 1; stern[n_] := stern[n] = Sum[stern[k], {k, n - Ceiling[(Sqrt[8*n -7]-1)/2], n-1}]; digNum[n_] := Length @ IntegerDigits[n]; digCount = 0; sum = 0; cumsum = {}; Do[s = stern[n]; If[digNum[s] > digCount, digCount++; AppendTo[cumsum, sum]]; sum += s, {n, 1, 25}]; Differences[cumsum] (* Amiram Eldar, Nov 30 2019 *)

More terms from Amiram Eldar, Nov 30 2019

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A066777 First differences of Stern's sequence A005230.

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A096858 Triangle read by rows in which row n gives the n-set obtained as the differences {b(n)-b(n-i), 0 <= i <= n-1}, where b() = A005318().

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A023536 Convolution of natural numbers with A023532.

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A225178 The generalized Conway-Guy sequence d_2(n).

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A014172 Apply partial sum operator twice to Stern's sequence.

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A014173 Apply partial sum operator thrice to Stern's sequence.

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A014175 Apply partial sum operator 4 times to Stern's sequence.

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A132156 Sum of all n-digit Stern numbers.

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